Quantum-inspired space search algorithm (QSSA) for global numerical optimization

This work presents the evolutionary quantum-inspired space search algorithm (QSSA) for solving numerical optimization problems. In the proposed algorithm, the feasible solution space is decomposed into regions in terms of quantum representation. As the search progresses from one generation to the next, the quantum bits evolve gradually to increase the probability of selecting the regions that render good fitness values. Through the inherent probabilistic mechanism, the QSSA initially behaves as a global search algorithm and gradually evolves into a local search algorithm, yielding a good balance between exploration and exploitation. To prevent a premature convergence and to speed up the overall search speed, an overlapping strategy is also proposed. The QSSA is applied to a series of numerical optimization problems. The experiments show that the results obtained by the QSSA are quite competitive compared to those obtained using state-of-the-art IPOP-CMA-ES and QEA.     

Convergence rates of a global optimization algorithm

Second order conditions for the (pseudo-) convexity of a function restricted to an affine subspace are obtained by extending those already known for functions on ⁿ . These results are then used to analyse the (pseudo-) convexity of potential functions of the type introduced by Karmarkar.This research was completed while the first author was on sabbatical leave at the Département d'Informatiques et de Recherche Opérationelle, Université de Montréal, and supported by NSERC (grant Q015807). This research was also supported by NSERC (grants A8312 and A5408) and la Coopération franco-québécoise (project 20-02-13).     

A counterexample to a global optimization algorithm

A counterexample to an algorithm of Dien (1988) for solving a minimization problem with a quasiconcave objective function and both linear and a reverse-convex constraint shows that this algorithm needn't lead to a solution of the given problem.     

Multi-objective optimization of parallel manipulators using a game algorithm

In this paper, a multi-objective optimization game algorithm (MOOGA) for parallel manipulators (PMs) is proposed. The proposed algorithm considers the volume of the regular cylindrical workspace, motion/force transmission performance, and stiffness performance as objective functions. First, the distributions of the objective functions in the complete parameter space are calculated and sorted by importance. Second, game weighting factors and lower bound values are assigned to different objective functions according to the engineering requirements. Finally, after multiple rounds of gaming according to the weighting factors and lower bound values, the objective functions reach an optimal balance point and obtain a balance intersection subspace. In addition, a new comprehensive stiffness index (CSI) is proposed, that takes the coupling of the non-diagonal elements into consideration. This index decouples the linear and angular stiffness and has definite physical dimensions as well as a clear physical meaning. A Lagrangian function is used to obtain the maximum and minimum stiffnesses at a given position along with their corresponding directions. To compare the difference between the CSI and the principal diagonal stiffness index (PDSI), a divergence index κ is proposed. 2UPR–RPU and 2UPR–2RPU PMs are employed as examples to implement the proposed algorithm, where U, P and R denote a universal joint, prismatic pair and revolute pair, respectively. The corresponding slice distributions of the local CSI and κ in the regular cylindrical workspace are presented. Additionally, the distributions of the extreme linear stiffness indices and their corresponding directions are presented. The results show that the CSI is decreased by 99% relative to the PDSI. The numerical results demonstrate the effectiveness of the algorithm proposed in this paper.     

Balancing global and local search in parallel efficient global optimization algorithms

Most parallel efficient global optimization (EGO) algorithms focus only on the parallel architectures for producing multiple updating points, but give few attention to the balance between the global search (i.e., sampling in different areas of the search space) and local search (i.e., sampling more intensely in one promising area of the search space) of the updating points. In this study, a novel approach is proposed to apply this idea to further accelerate the search of parallel EGO algorithms. In each cycle of the proposed algorithm, all local maxima of expected improvement (EI) function are identified by a multi-modal optimization algorithm. Then the local EI maxima with value greater than a threshold are selected and candidates are sampled around these selected EI maxima. The results of numerical experiments show that, although the proposed parallel EGO algorithm needs more evaluations to find the optimum compared to the standard EGO algorithm, it is able to reduce the optimization cycles. Moreover, the proposed parallel EGO algorithm gains better results in terms of both number of cycles and evaluations compared to a state-of-the-art parallel EGO algorithm over six test problems.     

A partition-based global optimization algorithm

This paper is devoted to the study of partition-based deterministic algorithms for global optimization of Lipschitz-continuous functions without requiring knowledge of the Lipschitz constant. First we introduce a general scheme of a partition-based algorithm. Then, we focus on the selection strategy in such a way to exploit the information on the objective function. We propose two strategies. The first one is based on the knowledge of the global optimum value of the objective function. In this case the selection strategy is able to guarantee convergence of every infinite sequence of trial points to global minimum points. The second one does not require any a priori knowledge on the objective function and tries to exploit information on the objective function gathered during progress of the algorithm. In this case, from a theoretical point of view, we can guarantee the so-called every-where dense convergence of the algorithm.     

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an \(\varepsilon \)-global minimizer is proved. At each iteration k, the framework requires the \(\varepsilon ^{(k)}\)-global minimizer of a subproblem, where \(\varepsilon ^{(k)} \rightarrow \varepsilon \). We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an \(\varepsilon ^{(k)}\)-global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the \(\varepsilon ^{(k)}\)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.     

Approximation algorithm for a class of global optimization problems

In this paper we develop and derive the computational cost of an ${\varepsilon}$ -approximation algorithm for a class of global optimization problems, where a suitably defined composition of some ratio functions is minimized over a convex set. The result extends a previous one about a class of Linear Fractional/Multiplicative problems.     

A parallel global optimization method and its implementation on a transputer system

This paper deals with a new parallel method for solving one-dimensional global optimization problems. We present the formulation of the decision rules of this method, the conditions for its nonredundant parallelization, the generalization for solving multi-dimensional problems and its implementation on a transputer system. There is also an account of some numerical experiments.     

A new parallel method for verified global optimization

We present a new parallel method for verified global optimization, using challenge leadership for the dynamic load balancing. The new approach combines advantages of two previous models: the centralized mediator model (see [1]) and the processor farm (see [2]). It has the following properties: centralization of the process; reduction of the number of box exchanges, communications used to send boxes from one processor to another; handling of the box that most probably contains the global minimizer. Numerical results show the efficiency of this method.     

SPT: a stochastic tunneling algorithm for global optimization

A stochastic approach to solving unconstrained continuous-function global optimization problems is presented. It builds on the tunneling approach to deterministic optimization presented by Barhen and co-workers (Bahren and Protopopescu, in: State of the Art in Global Optimization, Kluwer, 1996; Barhen et al., Floudas and Pardalos (eds.), TRUST: a deterministic algorithm for global optimization, 1997) by combining a series of local descents with stochastic searches. The method uses a rejection-based stochastic procedure to locate new local minima descent regions and a fixed Lipschitz-like constant to reject unpromising regions in the search space, thereby increasing the efficiency of the tunneling process. The algorithm is easily implemented in low-dimensional problems and scales easily to large problems. It is less effective without further heuristics in these latter cases, however. Several improvements to the basic algorithm which make use of approximate estimates of the algorithms parameters for implementation in high-dimensional problems are also discussed. Benchmark results are presented, which show that the algorithm is competitive with the best previously reported global optimization techniques. A successful application of the approach to a large-scale seismology problem of substantial computational complexity using a low-dimensional approximation scheme is also reported.     

A Bayesian algorithm for global optimization

A crucial step in global optimization algorithms based on random sampling in the search domain is decision about the achievement of a prescribed accuracy. In order to overcome the difficulties related to such a decision, the Bayesian Nonparametric Approach has been introduced. The aim of this paper is to show the effectiveness of the approach when an ad hoc clustering technique is used for obtaining promising starting points for a local search algorithm. Several test problems are considered.     

A deterministic algorithm for global optimization

We present an algorithm for finding the global maximum of a multimodal, multivariate function for which derivatives are available. The algorithm assumes a bound on the second derivatives of the function and uses this to construct an upper envelope. Successive function evaluations lower this envelope until the value of the global maximum is known to the required degree of accuracy. The algorithm has been implemented in RATFOR and execution times for standard test functions are presented at the end of the paper.Partially supported by NSF DMS-8718362.     

The extrapolated interval global optimization algorithm

This paper presents a new approach based on extrapolation to accelerate the linear convergence process of Vectorized Moore–Skelboe (VMS) algorithm. The VMS is a modified version of basic Moore–Skelboe (MS) algorithm, where the vectorization is used as a means to speed up the basic MS algorithm. We propose to further accelerate the converging process of VMS from linear to quadratic by combining the Richardson extrapolation technique with VMS. The effectiveness of the proposed algorithm is tested on various multivariate examples and compared with the unaccelerated conventional method, i.e., MS and well-known optimization software GlobSol. The test results show that the proposed extrapolation-based VMS offer considerable speed improvements over both the existing algorithms.     

Consistency of a myopic Bayesian algorithm for one-dimensional global optimization

A sequential Bayesian method for finding the maximum of a function based on myopically minimizing the expected dispersion of conditional probabilities is described. It is shown by example that an algorithm that generates a dense set of observations need not converge to the correct answer for some priors on continuous functions on the unit interval. For the Brownian motion prior the myopic algorithm is consistent; for any continuous function, the conditional probabilities converge weakly to a point mass at the true maximum.     

Local and global optimization by parallel algorithms for MIMD systems

This paper describes an implementation on the Neptune system at Loughborough University of Sutti's parallel (MIMD) algorithm [1–3] and an analysis of its performance. Parallel asynchronous versions of Powell's method [6] and Price's algorithm [7] are proposed, designed for efficient implementation on MIMD systems.This work has been developed during the author's stay at the Numerical Optimization Centre, Hatfield Polytechnic, England.     

Chaotic catfish particle swarm optimization for solving global numerical optimization problems

Chaotic catfish particle swarm optimization (C-CatfishPSO) is a novel optimization algorithm proposed in this paper. C-CatfishPSO introduces chaotic maps into catfish particle swarm optimization (CatfishPSO), which increase the search capability of CatfishPSO via the chaos approach. Simple CatfishPSO relies on the incorporation of catfish particles into particle swarm optimization (PSO). The introduced catfish particles improve the performance of PSO considerably. Unlike other ordinary particles, the catfish particles initialize a new search from extreme points of the search space when the gbest fitness value (global optimum at each iteration) has not changed for a certain number of consecutive iterations. This results in further opportunities of finding better solutions for the swarm by guiding the entire swarm to promising new regions of the search space and accelerating the search. The introduced chaotic maps strengthen the solution quality of PSO and CatfishPSO significantly. The resulting improved PSO and CatfishPSO are called chaotic PSO (C-PSO) and chaotic CatfishPSO (C-CatfishPSO), respectively. PSO, C-PSO, CatfishPSO, C-CatfishPSO, as well as other advanced PSO procedures from the literature were extensively compared on several benchmark test functions. Statistical analysis of the experimental results indicate that the performance of C-CatfishPSO is better than the performance of PSO, C-PSO, CatfishPSO and that C-CatfishPSO is also superior to advanced PSO methods from the literature.     

A perturb biogeography based optimization with mutation for global numerical optimization

Biogeography based optimization (BBO) is a new evolutionary optimization algorithm based on the science of biogeography for global optimization. We propose three extensions to BBO. First, we propose a new migration operation based sinusoidal migration model called perturb migration, which is a generalization of the standard BBO migration operator. Then, the Gaussian mutation operator is integrated into perturb biogeography based optimization (PBBO) to enhance its exploration ability and to improve the diversity of population. Experiments have been conducted on 23 benchmark problems of a wide range of dimensions and diverse complexities. Simulation results and comparisons demonstrate the proposed PBBO algorithm using sinusoidal migration model is better, or at least comparable to, the RCBBO based linear model, RCBBO-G, RCBBO-L and evolutionary algorithms from literature when considering the quality of the solutions obtained.     

An adaptive search algorithm for numerical optimization

The flexible polyhedron (simplex) search algorithm is reviewed and some of its shortcomings highlighted. Particularly, the fixed search parameters are shown to be a sure liability and an improvement is proposed. A unidirectional optimal search algorithm is substituted for the set of fixed rules usually employed to modify the simplex. This modification proves especially effective in dealing with “narrow valley” situations, normally encountered whenever the decision variables exhibit some degree of correlation. The new adaptive algorithm compares well with the parent simplex method, featuring less function evaluations and better convergence properties in cases where the classical search techniques perform poorly or fail altogether.     

A global best artificial bee colony algorithm for global optimization

The artificial bee colony (ABC) algorithm is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in the ABC algorithm regarding its solution search equation, which is good at exploration but poor at exploitation. Inspired by differential evolution (DE), we propose a modified ABC algorithm (denoted as ABC/best), which is based on that each bee searches only around the best solution of the previous iteration in order to improve the exploitation. In addition, to enhance the global convergence, when producing the initial population and scout bees, both chaotic systems and opposition-based learning method are employed. Experiments are conducted on a set of 26 benchmark functions. The results demonstrate good performance of ABC/best in solving complex numerical optimization problems when compared with two ABC based algorithms.